Empirical equivalence of different models of the general theory of relativity (GTR). This explanation of the old and new gravitational phenomena has assumed that nothing exists but two, opposite kinds of substances enduring through time. But the capacity of spatiomaterialism to explain those phenomena does not necessarily mean that it is equivalent to a single model of GTR on the received geometrical interpretation (which explains gravitation as a curvature in four dimensional spacetime). Thus,  it remains to be seen why there are infinitely many different, observationally equivalent models of GTR for any particular universe, or why “general relativity” (in one sense) seems to be true.

Since this explanation of gravitation is based on the spatiomaterialist explanation of the truth of STR, space is assumed to be a substance, and we are liberty to take as our reference frame the inertial frame at rest in space outside of gravitational influences where the one-way velocity of light is the same both ways in every direction. That inertial frame is at rest relative to the inherent motion in space, and the inherent motion itself is at rest relative to space (in other words, that inertial frame is at rest relative to the ether, which, in turn, is at rest relative to space). The times and places of events assigned by observers on such an “absolute reference frame” would be accurate, because his clocks would not by mis-synchronized

Consider a gravitational field imposed by a gravitating body of some kind. It will be accelerating the inherent motion (or ether) toward itself according to the inverse square law. Of all the reference frames that would be accelerated toward the gravitating body, the one with the most accurate times and places of events would be the one that is at rest relative to the inherent motion itself (or the ether) as it is being accelerated toward the center of gravitation. To be sure, such a reference frame could not have clocks synchronized everywhere, since any large rigid object would be torn apart by the difference in forces acting at different points. But if observers on that reference frame could use GTR (or this ontological explanation of the gravitational force) together with light signals received from other objects to figure out where and when events occur throughout the gravitational field. That is, they would determine the “simultaneity hypersurface in curved spacetime” from their reference frame, and since that would correspond to what is really happening to substances at that moment as they endure through time, their reference frame can be called the “absolute model” for GTR, by analogy to the inertial frame of the absolute observer in STR.

The reason that there are many different empirically equivalent models for any such situation is that there are other reference frames which differ from the absolute reference frame only by have a velocity relative to the inherent motion itself that is being accelerated inward. They are empirically equivalent locally, because they suffer Lorentz distortions that mask their velocity relative to the inherent motion. And observers on each of them could use GTR together with information received from events elsewhere to determine their “simultaneity hypersurface in curved spacetime.” They would all disagree with one another, like different inertial observers outside gravitation, and there would be no way for them to tell by experiment which reference frame was the absolute reference frame.

That is, each possible model of GTR is adapted to the trajectory of one of the many different particles that could be in inertial motion at any point, and their different velocities relative to the inherent motion would give them, incipiently, at least, different standards of simultaneity (that is, each determines a different “simultaneity hypersurface in curved spacetime”). Any pair of such reference frames may have a high velocity relative to one another as they pass one another at that point, but each would observe Lorentz distortions occurring in the other reference frame, and thus, the symmetry between them would make it impossible to for them to tell which reference frame is at rest relative to the inherent motion in space that is being accelerated toward the center of gravitation.

This explains why models based on different reference frames are empirically equivalent as far as different velocities relative to the inherent motion is concerned. But neither can anything known about the effects of the gravitational force be used to distinguish one reference frame from another. Even of observers on the reference frames accepted the spatiomaterialist explanation of the nature of gravitation as an acceleration of the inherent motion by the gravitating body, that would not single out the absolute reference frame from the rest. (Or if the observers think in terms of GTR and see gravitation as a “curvature of spacetime” caused by gravitating bodies, that does not compromise their empirical equivalence.)

The absolute model cannot be assumed to be the one based on the local inertial frame that would result from accelerating all the way in from being at rest outside the gravitational field, for the gravitating body may itself have a non-zero velocity relative to the inherent motion in unstressed space.

It might seem possible to measure an object’s velocity relative to the inherent motion by using the gravitational time dilation of objects at rest in the gravitational field to determine their velocity relative to the inherent motion. But that will detect only the increase in the velocity of the inherent motion as a result of being accelerated toward the center of gravity to that point from outside the gravitational field, and that will not determine whether the gravitational field itself is in motion relative to the inherent motion outside gravitation.

Or it would be possible, in principle, to use the difference between light signals and gravitational signals to detect absolute rest, if gravitational forces propagated at a different velocity from light. But since the force that accelerates the inherent motion in space propagates through the inherent motion at the same velocity as light, its effects are explained equivalently by each model in the same way as light.

The equivalence of inertial frames that Einstein meant by “general relativity” can be explained, therefore, by special relativity. That is, the empirical equivalence of different models of GTR can be explained as the empirical equivalence of local inertial reference frames that have different constant velocities relative to the accelerating inherent motion. There is no way to determine which of their standards of simultaneity is correct, for there is no way to detect rest relative to the inherent motion. And none of the interactions between space and matter that constitute the force of gravity betrays which reference frame is the absolute model.

Though gravitation just happens to work in such a way that absolute rest relative to the inherent motion cannot be detected, the fact that it works that way could explain why Einstein was able to deduce his law describing the unexpected effects of gravitation from the assumption that all different local inertial frames are equivalent, or “general” relativity.

The spatiomaterialist explanation of gravitation has been presented as an ontological explanation of the truth of Einstein’s general theory of relativity. Since what is crucial to such an ontological explanation is identifying the aspects of the substances constituting the world to which the theory corresponds, I have presented only a qualitative argument. I have shown how GTR could be true, even if nothing existed but substances enduring through time, and every possible photon has a determinate location and velocity in absolute space at each moment as it is present (because the inherent motion itself is accelerated and, thus, moving through space). Though I have said enough about the quantitative factors to make clear how it would predict the same quantitatively precise measurements, I have not shown mathematically that it is equivalent.

That is an exercise I leave up to mathematically inclined readers. It affords an opportunity to refute ontological philosophy, for if it can be shown that there is no way that the acceleration of the inherent motion in space can yield the quantitatively correct predictions for all the relevant phenomena, we will have defaulted on the mortgage we took out to use spatiomaterialism as the foundation for the necessary truths of ontological philosophy, and the project will have failed. I see not reason to belief that that can be done. But like any basically empirical argument, ontological philosophy is vulnerable to empirical falsification, and thus, it must stand up to such challenges.

We can also see, at this point, why philosophers of science have not recognized the superiority of substantivalism about space to substantivalism about their spacetime. Instead of inferring to the best ontological explanation of everything in nature, they have let their ontology be determined by realism about the highly mathematical theories that physics has accepted as the best efficient-cause explanation of what happens in nature. Philosophers of spacetime simply assume that every theory about space and time, including Newton’s (and, thus, spatiomaterialism), can be represented as just another variety of spacetime theory using differential geometry.^[1]

What spatiomaterialism offers, however, is a different kind of model of GTR. It explains ontologically why Einstein’s field equations are true by showing how gravitational phenomena can be constituted by space and matter as substances that exist only at the present moment. To treat spatiomaterialism as the belief in a “simultaneity hypersurface in a four dimensional spacetime manifold” is to abstract from such basic ontological issues as the nature of existence and time and to judge these theories only as efficient-cause explanations, that is, by their predictions of precise measurements.^[2]

And when we judge all these theories by their capacity, as ontological theories, to account for everything observable about the world, including real change, the empirical superiority of an ontology of enduring substances is obvious, as we have seen, because of its explanation of the nature of time and existence.

Spacetime, whether curved or flat, cannot explain how the present is different from the past and the future, because spacetime cannot be a substance enduring through time as long as time is part of its structure. Thus, neither can it explain real change, because nothing ever comes into existence as time passes nor goes out of existence. (And as we have seen, attempts to avoid falsification by our experience of real change by adding subjective substances to the ontology makes it more complex encounters problems relating eternal and enduring substances as a single world, and is in any case ad hoc.)

Spatiomaterialism differs ontologically from Einstein’s GTR in just the way required to explain real change. Though it explains gravitation in much the way Einstein proposed — as an effect of the container of material objects on the path they follow — it replaces curved spacetime with an acceleration of the inherent motion itself. Since that is nothing but an aspect of space and matter as substances enduring through time, given how they are related, it explain why the present is different from the past and the future and “real change” is ontologically possible.